IL NUOV0 CIMENT0 VOL. 4A, N. 3 1 Agosto 1971

Second.Class Vector Currents and Scalar Mesons (*)C).

S. ELIEZEg and 1 ). SINGER

Department o] Physics, Technion-Israel Institute o] Technology - Hai]a

(ricevuto il 20 Gennaio 1971)

S u m m a r y . - - The nonconscrved part of the vector currents, assumed to have second-class transformation properties, is taken to be propor- tional to interpolating scalar fields. By this procedure, new sum rules for second-class form factors are obtained, which arc then used to calculate the ~] --> r: and E-+ A semi-lcptonic appropriate transitions. The couplings of scalar mesons to baryons are shown to be two orders of magnitude smaller than those of pseudoscalar mesons.

1 . - I n t r o d u c t i o n .

More t h a n ten years ago WEINBEgG (1) classified the s t rangeness -conserv ing

weak hadron ic cur ren ts accord ing to the i r t r a n s f o r m a t i o n proper t ies u n d e r the

G-par i ty opera tor . V ' ( A ~) is a first-class cu r r en t if

(1) G I V ' ( 5 S = O)G71 = V ' ( A S = O), G ~ A ' ( A S = 0 ) G ~ I = - - A t ' ( A S = O),

where V ~ (A ~) are vec to r (axial-vector) currents . W e use the symbol G~ for

the usua l G-par i ty operator ' : G ~ C exp [iJrI2] , C being the charge conjuga-

t ion opera tor , and I t he isotopic spin. Second-class cur ren ts t r a n s f o r m u n d e r G~

wi th oppos i te signs f r o m those g iven in (1).

This classification has been ex tended to the s t rangeness -chang ing cur ren ts b y

WOLFENSTEIS" (~), who i n t r o d u c e d a new ope ra to r Gv, where G v =~: C exp [i7~V~],

(*) To speed up publication, the authors of this paper have agreed to not receive the proofs for correction. (**) Research supported in part by Stiftung Volkswagenwerk. (1) S. W]~INBI~RG: Phys. gev., 112, 1375 (1958). (2) L. WOLF]~NSTt~IN: Phys. Rev., 135, B 1436 (1964).

638

SECOND-CLASS VECTOR CURRENTS AND SCALAR MESONS ~ 3 ~

V2 being the second c o m p o n e n t of the V spin U). I n ana logy wi th eq. (1) one

now has

(2) GvV~(AS = 1)G~ ~ = ~ V ' (AS = 1 ) , GvA ' (AS ~- 1)G~ ~ = - - ~ A ' ( A S = 1 ) ,

wi th ~ = ÷ 1 for first-class currents , and ~?= - - 1 for the second-class ones (4).

I n the Cabibbo SU3 mode l of weak in te rac t ions (5), in which the s t range-

ness-conserv ing cur ren ts (AS = 0) and the s t rangeness -changing (AS = 1) cur-

ren ts be long to the same octet , eq. (2) follows (~) as a consequence of eq. (1).

The p resen t t h e o r y of weak in te rac t ions assumes the existence of first-class

cur ren ts only, b u t in our p resen t s ta te of knowledge the re is no evidence agains t

t he exis tence of the second-class ones (7). Moreover , when the SUa s y m m e t r y

is b roken , second-class ma t r i x e lements are possible (s).

W e as sume in this work t h a t the had ron ic vec to r cu r ren t can be wr i t t en

as the sum of first a nd second class currents , V , = V ~ ) ÷ V~ ~). The first-class

cu r ren t is conserved, while for the second-class cu r ren t we assume t h a t its

d ivergence does no t vanish . W e fu r the r p roceed b y using the not ion of par t ia l

conse rva t ion of the vec to r cu r ren t (PCVC), so t h a t the d ivergence of the cur-

r en t is p ropo r t i ona l to an in t e rpo la t ing scalar field (9). I n our previous work (9),

we inves t iga t ed t he impl ica t ions of this a s s u m p t i o n for the s t rangeness-con-

serving vec to r cur ren t and we der ived (( Goldberger -Tre iman- l ike ~ relat ions,

which were found to be cons is tent wi th w h a t e v e r exper imen ta l or inferred

knowledge exists a b o u t A a nd ~ lep tonic decays . I n this pape r we ex t end our

discussion to the honer of vec to r currents . Hence fo r th , we shall assume the

divergences of vec to r cur ren ts to be p ropo r t i ona l to a honer (or possibly octet) of scalar mesons (~o)

(3) ~. Va" = qom~o~o,

(3) For the I , U, V spill notation see C. A. LEVlNSON, H. J. LIPKIN and S. -~[ESHKOV: NUOVO Cimento, 23, 236 (1962); Phys. Rev. Lett., 10, 361 (1963). (4) As is well known, nonstrangc mesons are eigenstates of the G 1 operator. Appro- priately, eigenstates of the G v operator can be formed. For instance, the triplet V-spin mesonic states (K-, ½~°+(~/3/2)~1, K +) have G v - -1 and the V-singlet state ((~/3/2)~°--½r;) has Gv= + 1. Matrix elements of first (second) class vector currents between states with definite G-parity (Gx or Gv) vanish if the G-parities of initial and final states are different (alike). For the axial-vector era'rents the opposite situation occurs. (5) N. CAraBao: Phys. Rev. Lett., 10, 531 (1963). (6) I. BENDER, V. LI:NKE and H. J. ROTHE: Nuel. Phys., 9B, 141 (1969). (7) L. WOLFENSTEIN: Proceedings o] the Heidelberg International Con/erence on Ele- mentary Pa~'ticles (Amsterdam, 1968), p. 289. (s) M. ADEMOLLO and R. GATTO: Phys. Bey. Lett., 13, 264 (1964). (9) S. ELIEZER and P. SINGER: :Yucl. Phys., 11 B, 514 (1969). 0 °) The experilnental situation concerning the existence of scalar particles is still

640 S. ]~LIEZER and P. SINGER

where a is an S U3 index. The t ransformation properties of V~ 2) imply that the

octet of scalar mesons has the same charge conjugation as the pseudoscalar

octet. ]~ is defined by

(4) (2~) s ~/2~q°<a(q)] V$]0> = - - ]~q,,

and ] = - - 1 ~ follows from the charge conjugation t ransformation properties

of second-class currents.

In Sect. 2 we investigate the s tructure of vector matr ix elements and obtain

formulae relating the strong couplings of scalar mesons to form factors of vector

transitions among mesonic nnd baryonie states. In Sect. 3 these formulae

are used to derive sum-rules for second-class form factors. I n addit ion to the

Ademollo and Gatto sum rules (8.~), we obtain three new sum rules. These

sum rules are then used for calculating semi-leptonic processes. I n the last

Section we estimate strong couplings of scalar mesons and discuss some impli-

cations of our results.

2. - Structure of vector matrix elements and their connection with the strong couplings of scalar particles.

We now proceed by considering matr ix elements of the vector currents

between baryonic states. The case of lambda and proton states is worked out in detail to exemplify our procedure; for this part icular example eq. (3)

is used with a ×+ scalar particle, with quan tum numbers I a~ = 1-. The general

matr ix element of the vector current between A and p states is given by

(5) (27~)8 ( ~ ) ~ ( p ( p 2 ) J V~+~5(x)JA(p,) } =

= U~(P2) {]~'~(q2)[y~ 4- (m.--mA) ~] +

4-/~.,(q2) -~ 4- ]~.~(q~)a,~q, UA(p,) exp [iqx] , q" ~ ( P l - P2) ~,

where y" and (r#'q~ are first-class terms, while q" is a second-class term. This

unsettled. The indications are that a scalar octet (or nonet)with Inass around 1 GeV does indeed emerge from the existing data. In particular the I = l (rCN) and the I = 1(×) components have been more often reported (11.12). (11) PARTICLE DATA GROUP: Rev. Mod. Phys., 41, 109 (1969). (12) W. G. TRIPPE, C. Y. CHIEN, E. MALAMUD, J. I~/[ELLEMA, P. E. SHLEIN, W. E. SLATER, D. H. SwoRN and H. K. TIcgo: Phys. Lett., 28B, 203 (1968). (13) R. GATTO: in Symmetries in Elementary Particles, edited by A. ZICHICHI (New York, 1965), p. 175.

SECOND-CLASS VECTOR C U R R E N T S A N D SCALAR MESONS 6 4 1

choice (14) of fo rm factors guarantees t ha t ]~(q2) and ]z(q~) pick up contr ibu-

tions f rom spin 1- and 0 + s ta tes respectively. Taking the divergence of eq. (5)

we get

o o ½ [ P~P2 \ < , i~,V~+MO)IA > =/~..(q2) U~ Vh

On the other hand, we t ake the ma t r ix e lement of eq. (3) (for a = x +) be tween

IA(p~)> and <P(P2)[ states, and using

(7)

we get

(s)

o o ) ½ PlP~

g×+~h(q 2) Up(p2) UA(p,) ~ (2~) ~ m ~ <p(p~)l(~ + m~)~×+lA(pi)>,

o o ½ 2 2

By compar ing eqs. (6) and (8), we obtMn

2 2 (9) a.p 2 g~+~A(q )/×m~

Is ( q ) - m~ - q~

F r o m eq. (3) and the ma t r ix e lement of the vector current be tween the baryons Bx and B2, we obta in in the same manner as above the following relat ion for the second-class fo rm factors /2n"S~(q~):

2 2 (10) p,.~,t~,2~ _ gsB~,(q )/sins

2 k~/ / 2 q2

where S denotes the scalar part icle with the app rox ima te q u a n t u m numbers

and gs~,~ are the coupling of baryons to scalar mesons.

I n eq. (5) a pole at q ' = 0 should not occur, since a charged part icle with zero mass is not known to exist, and one mus t therefore require t h a t

Ap 0 (11) 1~'~(0)- Is'( ) 7Vt A - - ~rytp

B y compar ing eq. (9) a t q " = 0 with eq. (11), we get

(12) 1#*(0) - g~+Ao(0)l~, 7it A - - ~7/p

(14) K. HEF~ and B. STECE: Zeits. Phys., 202, 514 (1967).

41 - I 1 N u o v o C i m e n t o A .

642 s. ELI~ZER and P. SINGER

or in general for the vector form factor ]~',B'(O) we have

(13) 1~, .~(0) - g ~ , ~ , ( o ) / ~ ml - - m 2

Next we turn to the mat r ix elements of the vector current between pseudo- scalar states ]Pl(ql} and (P~(q2)l, whose general form is

(14) (2Jr)*(2q°2qg)~(P=(q=)lV'lP~(ql)} ----

_ p,v,V~(q2)[(q 1+ t, q~ ] F~.P,~.=~ q~ - q~) - ~ ( ~ - m , =) + ~ , ~ , ~ ; q" - ( q l - q~)'.

Here (ql + q2)" is a first-class te rm and (ql - - q2) is a second-class one (14). The form factors are so chosen tha t in a dispersion relation 2~1(q 2) has contri- but ion f rom 1- states and F d q ~) f rom 0 + states only.

F r o m eqs. (3) and (14) we obtain the following relations:

9 n 2 2 (15) F~.~.e~.2~ ]s sgse, P~(q )

~ J - m ~ - - q ~ '

(16) P [ ' ~ ( 0 ) - / ~ e ~ . , ~ ( o ) ~ - - ~ •

In arriving at (16), we used again the argument concerning the lack of a pole at q2= 0, which gives

(17) F~"P~(0) = F~'~'(0) 2 2 "

m 1 - - m ~

gsp,P~ are the couplings of a scalar part icle to two pseudoscalars.

3. - Sum rules for second-class form factors and semi-leptonic processes.

As the exper imental informat ion concerning scalar mesons is still meagre,

it seems tha t the s traightforward thing to do at this stage is to assume an S U3-symmetric form for their interactions. Then the interact ion Hamil tonian

of a scalar octet to two pseudoscalar octets is

(18) Hsp P = g Tr (SP.P) ,

where the SU3 octets are represented by traceless 3 × 3 matrices S and P.

(15) N. CABIBBO: Phys. Rev. Lett., 14, 965 (1965).

S E C O N D - C L A S S V E C T O R C U R R E N T S A N D S C A L A R M E S O N S ~ 4 3

For the interactions of scalar mesons with baryons we accordingly take

(19) H s ~ = g 'Tr ([B, B]S) -F gD Tr ({B, B} S ) ,

m where B ~nd B are the SU3 matr ix representat ion for the baryons.

Using now eqs. (10) with (19) as well as (15) with (18) we obtain the fol-

lowing sum rules for second-class form factors:

(20)

(21)

(22)

(23)

(24)

(25)

(26)

][-.A(q~) p-Ap~,

7: A , v 2 -"~o ~+ 2 ~/~ ~t~ (q) +/~-.A(q~)] ~ - . - - = 1 2 ' ( q ) + J T " ( q ) , n,p 2 ~ ' - ~'0 2

]3 (q) = --1~" - ( q ) ,

1 ] / 5 ~,- h 2 l~."(q') = ~ l~-.~°(q ') + -~ l , • (q ) ,

V/:2F~-"*'(q 2) = F~u'"~+(q2) ,

~o.~(q~) = _ ~ . , : + ( q ~ ) ,

1 / PY(q~) = ~1/~ F[°'~+(q~) •

Equat ions (20), (21) were previously derived by ADE~OLL0 and GATe0 (8), by assuming tha t the weak vector currents and the electromagnetic current belong to the same uni ta ry octet, and the breaking of the symmet ry is due to a t e rm behaving like the eighth component of an octet. Equat ions (24) and (25) are likewise implied in the general formula obtained by GATTO (12) for a second-

class t ransi t ion of pseudoscMar mesons, with the same assumptions as in ref. (8). The eqs. (22), (23), (26) are new and are not obtainable within the symmet ry requirements imposed by ADE~OLL0 and GATT0 in whose t r ea tmen t all the strangeness-conserving form factors ]~,,~(q2) (except for /~.A(q~)) and /~P"P'(q") vanish. We shall umplify this point in the last Section.

Using our new sum rules, we c~n c~lculate the form factors appearing in

the semi-leptonic decays of ~ and E. The process ~-->~±l~v is of part icular theoret ical interest , as i t represents a unique example of a pure second-class t ransi t ion (see ref. (16) for a detailed discussion).

(16) p. SINGER: Phys. Rev., 139, B 483 (1965); L. B. 0KVN' and I. S. TSUKERMAN: 2urn. Eksp. Teor. Fiz., 47, 349 (1964) (English translation, Soy. Phys. JETP, 20, 232 (1965)).

644 S. :ELIEZER o~n(t p . SINGER

Using now eq. (26) a t q2= 0 as well as eq. (17) for the ~-+rc ± and

K ° - + K + semi-leptonic transi t ions, we get

(27) / ~ . ~ ( 0 ) = m ~ . - - ~rt~+ m2 m2 F~K°.K+(0) = 1.1-10 - 2 ,

~ - - re+

where we used the SU2 value for FX°'x+c0~ This value implies an ~ par t ia l decay width

(28) F(? -+ rdv) _ 2.1.10 - '4 F(~ ~ all)

where F(~ --> =iv) includes all possible leptonic modes (i.e. ~= e ~, ~*) and we

used F(~ --~ all) : 2.6 keV. The value ob ta ined in (28) agrees with our previous es t imate (9). This is interest ing, in view of the fact t ha t in the earlier c~lcula-

t ion we depended on the values of the mass and decay width of 7:~, while rela-

t ion (27) requires the use of well-determined quanti t ies only.

Another AS : 0 t rans i t ion where second-class currents can p lay an im- po r t an t role, and which we can calculate with our new sum rule (23), is the Y±-~Ae±~ decay. F r o m eq. (23) and the relations be tween ]~,.B, and ]~,s,

required b y the lack of a pole a t q2= 0 (e.g. eq. (11)) we get

(29) /~;.h(0) = ~= (m,--mp)/~'P(O)--(l/~)(mr..--mso)i~ "(0) m ~ mA

By using J~'~(0)= 1, /~- '~'°(0)= v ~ , J ~ ° ' ~ ÷ ( 0 ) = - v / 2 , one obtains

( 3 0 a ) / ~ - ' A ( 0 ) = - - 3 . 4 1 " 1 0 - 3 ,

( 3 0 b ) ]~+.A(O) = - - 1 . 8 4 " 1 0 - 2 .

The fo rm factors ]1(0) were assumed to be unrenormal ized to first order in sym-

m e t r y breaking according to the Ademol lo-Gat to theorem (s,1D.

4 . - D i s c u s s i o n .

4"1. - Before discussing some general features of our model, we should like to present results which are of relevance to the strong couphngs of scalar

m e s o n s .

(17) S. FUmNI an6 G. FURLAN: Physics, 1, 229 (1965).

SECOND-CLASS VECTOR CURRENTS AND SCALAR MESONS 6 ~

Using eq. (13) for n - + p and E ° - + E + transit ions, as well as the SU3 val-

ues for the g~;,~(O) and g~,=or-(O) couplings, we obtain

(31) ] ~ , _ m . - - m , ]~'P(0) ~0.8.10_~, i,,÷ m_~,- m~÷ 1~'.~(0)

where the equal i ty sign would hold when using SU3 values for the ]1(0) fo rm

factors. I t should be noted t ha t b y choosing these transit ions, one can obvia te explicit knowledge of the F/D rat io a t this stage.

I n order to obta in an es t imate for f~, we can use our previously obtained (~) relat ions for the f=~, name ly

2 2 K ° K + (m~-- m~+)2'~ " (0)

grc#KK

Iden t i fy ing ~ with the particle of mass mn = a 0 1 6 M e V and width

F = 2 5 MeV, one obtains

(32) ] ~ = 2.5 M e V , ]~ = 312.5 M e V .

Hence, we obtain ~ rat io ],J]~= 2.2, which is on the higher side of the esti-

ma tes of var ious authors , ranging between 0.1 to 2. One should r emark however t ha t the value we obta ined for ]~ depends on the mass and width of ~z~v , and

change in these parameters , i.e. the doubling of ~ ' s width, would approxi- m a t e l y decrease ]~ b y a factor of ~/z.

Using now eq. (]3) for the t ransi t ions n - ~ p and A - * p , and the values given in (32), we obta in the values of the strong couplings of scalar mesons to ba ryons to be

(33) g~ = 0.45 , gD _ 0.22 . g~

These r e m a r k a b l y small values predic ted by our model for the couplings of scalar mesons to baryons are quite interest ing in view of the small cross-sec- tions observed for the product ion of ~z~ and × mesons in 7: p and K - p collisions.

4"2. - Our poin t of view in this work is t ha t the conserved pa r t of the vector

current has first-class t rans format ion propert ies , while the noneonserved pa r t

is of second-class type. I n this way, the existence of second-class current m a y be the source of the S U3 breaking.

Fu r the r on we obta in various results for these second-class fo rm factors

by our addi t ional assumpt ion of the scalar-meson dominance of the divergences of the second-class currents. I t should be poin ted out t ha t b y the PCVC

assumpt ion we lump together e lectromagnet ic and SU3 breaking effects, as bo th thus cont r ibu te in an inseparable way to the divergences of the currents.

646 S. ELIEZER and P. SINGER

I n th i s sense, we c a n n o t e x p l i c i t l y s e p a r a t e t h e e l e c t r o m a g n e t i c a n d t h e S U3-

b r e a k i n g s t r o n g c o n t r i b u t i o n s to m a s s d i f fe rences a n d f o r m fac to r s . H o w e v e r ,

i t shou ld b e s t r e s sed t h a t p r e v i o u s r e su l t s , such as t h e s u m ru les of A d e m o l l o

a n d G a t t o (eqs. (20), (21), (24), (25)) r e m a i n u n c h a n g e d in our mode l . T h e

n e w re su l t s we o b t a i n (e.g. eqs. (27), (29)) c o n t a i n r e l a t i o n s b e t w e e n mass dif-

f e rences a n d f o r m fac to r s w h i c h a r e u n r e l a t e d in t h e c o n v e n t i o n a l ~pp roaehes .

4"3. - T h e a c t u a l p rocesses we c a l c u l a t e d in Sect . 3 a r e u n f o r t u n a t e l y dif-

f icu l t to m e a s u r e . T h e ~-->7~ev is s t i l l a f a r c ry f r o m p r a c t i c a l poss ib i l i t i e s ,

wh i l e for E - + A e v t h e p r e s e n t e x p e r i m e n t a l v a l u e (~8) is /~(0) = 0.7 q- 0.4,

wh ich is s t i l l t oo c r u d e to be c o m p a r e d w i t h our r e s u l t in eqs. (30).

O n t h e o t h e r h a n d , t h e r e is t h e h o p e t h a t t h e s e f o r m f a c to r s wi l l l e n d t h e m -

se lves to m e a s u r e m e n t in n e u t r i n o - i n d u c e d r eac t i ons . Hence , i t shou ld be

pos s ib l e to check in t h e f u t u r e t h e s u m ru les p r e s e n t e d in eqs. (20)-(26).

(18) C. B_&LTAY, 19. FRANZINI, :R. NEUMAN, H. NORTON, N. YE~, J. COLE, J. L~E- FRANZINI, R. LOVEr.ESS and J. 1V[CFADYEN: Phy8. Rev. Lett., 22, 615 (1969).

• R I A S S U N T O (*)

Si suppone c h e l a par te non eonservata delle eorrenti vettoriali , ehe si presume abbia propriet£ di trasformazione della seeonda classe, sia proporzionale ai campi sealari interpolant i . Con questo procedimento si ottengono nuove regole di somma per i fa t tor i di forma di seconda classe, che si usano poi per caleolare Ie appropr ia te t rasformazioni semfleptoniche ~-+= e Z ~ A . Si mostra che gli aceoppiamenti dei mesoni scalari ai barioni sono di due ordini di grandezza minori di quelli dei mesoni pseudoscalari.

(*) Traduzione a cura della Redazione.

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